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The Chain Rule
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d r
d
( x )  r  x r 1 is called the Power Rule, and recall that I said [(2 x  1)3 ] can’t be done by the power rule because the
dx
dx
base is an expression more complicated than x. In other words, in order to use the power rule, the base must be x, or the
variable you are differentiating with respect to. However, it doesn’t mean we can’t differentiate (2x + 1)3. All we need is
a rule called the Chain Rule, more appropriately, Chain Rule with the Power Rule (three versions are provided, it’s up to
you to choose the one you like):
d
d
( g ( x)) r  r  ( g ( x)) r 1  g ( x) 
1.
dx
dx
d
d
(base)r  r  (base)r 1  base
2.
dx
dx
d
d
3.
(expression)r  r  (expression)r 1  expression 
dx
dx






Here is how it works:
d
1.
[( 2 x  1)3 ] 
dx
2.


d
(3x 2  4 x  6)5 
dx
3. If f(x) = 3(6 – 5x2)4, find f (x).
Product Rule, Quotient Rule and Chain Rule (let’s throw them together)
Find f (x) for each of the following functions:
(4 x  5) 2
( x  3)( x  4)
2
3
1. f(x) = (x – 2) (x + 3)
2. f ( x) 
3. f ( x) 
x3
( x  2)2
Derivatives of Trigonometric Functions—Derivative of sin x
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If f(x) = sin x, what is f (x)?
Recall the limit definition of derivative: f ( x)  lim
h0
f ( x  h)  f ( x )
h
Chain Rule on sin (g(x)):
If f(x) = sin (g(x)), then f (x) = cos (g(x))g(x). That is, d/dx[sin (expression)] = cos (expression)d/dx(expression).
Examples:
For each of the following functions, find its derivative.
1. f(x) = sin x2  f (x) =
2. f(t) = sin [(t + 2)(3t2 – 4)] 
 x 
3. g(x) = 2 sin  2  
 x 1 
Derivatives of Trigonometric Functions—Derivative of cos x
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If f(x) =cos x, what is f (x)?
This time we are not going to use the limit definition to find f (x), but rather, recall cos x = sin (
):
Chain Rule on cos (g(x)):
If f(x) = cos (g(x)), then f (x) = ____________. That is, d/dx[cos (expression)] = _________________________.
Examples:
For each of the following functions, find its derivative.
1. f(x) = cos (x2 + 2x – 1)  f (x) =
2. f(t) = cos [(t – 3)(2t2 + 1)] 
 x 
3. g(x) =3 cos 2  
 x 1 
Derivatives of Trigonometric Functions—Derivatives of the Other Four
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If f(x) = tan x, what is f (x)?
If f(x) = cot x, what is f (x)?
If f(x) = sec x, what is f (x)?
If f(x) = csc x, what is f (x)?
Recall:
Recall:
Recall:
Recall:
Chain Rule on these functions:
If f(x) = tan (g(x)), then f (x) = ______________________ If f(x) = cot (g(x)), then f (x) = ______________________
If f(x) = sec (g(x)), then f (x) = ______________________ If f(x) = csc (g(x)), then f (x) = ______________________
Examples:
For each of the following functions, find its derivative.
1. f(x) = tan x2 sin 2x  f (x) =
2. f(t) = csc (t – 3) cot (2t2 + 1) 
3. g(x) =
sec(1  x)

cos( x 3  2)
Summary of Derivatives of All Six and Precaution on Chain Rules
The table on the right shows the derivatives of the six basic trig.
functions (notice the derivatives of the three cofunctions—cosine,
cotangent and cosecant—have a ________ sign). Of course,
Chain Rule can be applied to each one of them (see bottom table).
The order of applying Chain Rules
We have to apply Chain Rule for finding the derivative of many
functions, and for some of them, we need to apply Chain Rule
more than once, and the d
ORDER we apply the dx sin( g ( x)) 
Chain Rule MATTERS.
d
tan( g ( x)) 
dx
Examples:
Find the derivatives of d sec(g ( x)) 
dx
the following functions.
f(x) = sin x3 vs. f(x) = sin 3 x
f(x) = cot 2 [sin(x2 + 3)]
d
(sin x)  cos x
dx
d
(tan x)  sec2 x
dx
d
(sec x)  sec x tan x
dx
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d
(cos x)   sin x
dx
d
(cot x)   csc2 x
dx
d
(csc x)   csc x cot x
dx
d
cos(g ( x)) 
dx
d
cot(g ( x)) 
dx
d
csc(g ( x)) 
dx
f(x) = tan 2 (cos x) vs. f(x) = tan (cos 2 x) vs. f(x) = tan (cos x2)